In this section we introduce an alternative representation for a Green function based on Feynman path integrals. This approach is most suitable for the analysis of the wave propagation and scattering in a line-of-sight region. This section provides some basics for study in an unbounded medium, while later in Section 3*** we introduce a path integral representation in the presence of a boundary.
where:
- are the coordinates of the observation point and source, respectively;
- are the vectors in the plane orthogonal to the direction of propagation x, and;
- is the differential in the space of continuous trajectories.
The action S is given by:
where:
- is an unperturbed (when δε = 0) Lagrangian in the small-angle approximation;
- is an unperturbed potential;
- , the angle brackets < … > here and later denote averaging over the ensemble of dielectric permittivity;
- is a random component of
We can assume that the fluctuations in δε are statistically uniform:
and have different correlation scales:
- Lx in direction x and;
- L⟂ in the plane (γ, z).
where:
Introduce a two-dimensional spatial spectrum of the fluctuations de in a plane (γ, z):
Then for γS and Ds we obtain:
- θd, the gradient of the ray trajectory due to either the position of the correspondent or regular refraction (if any), in case of normal refraction θd is given by Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and Applicationsthis equation);
the angular size of the Fresnel zone;
- θs, the characteristic angle of scattering on the fluctuations δε, θs ~ 1/kL⟂.
Therefore θR = max{θd, θF, θs}.
and:
where:
and:
“Local” conditions of the Markov approximation are bounded by inequalities (3) and (4), for an average field we also need smallness of the attenuation of the average field over the distance of Lx:γS(Lx) ≪ 1. “Non-local“ conditions lead to the inequality:
where Rc is a coherence radius of the scattered field in a plane (γ, z), which depends on the distance x, Rc ≡ Rc(x). The parameter Rc can be found from:
The rigorous mathematical solution to the problem encountered difficulties in a general case of the multi-scale inhomogeneities of δε and, therefore, the basic physics has to be involved for a qualitative analysis.
We assume a small value of the mean-square variation of the fluctuations in a phase difference of the fields approaching any points separated in space by a distance of the order of the Fresnel zone size. Hence, this assumption is rendered by the inequality:
Such a straight-line approximation of trajectories in a path integral is similar to the extended Huygens–Fresnel principle. The applicability of extended Huygens–Fresnel principle shown that this approximation besides yielding the exact solution for the first two moments, provides a qualitatively correct solution for the high-order moments of the wave field.
Numerical Methods of Parabolic Equations
Among many numerical methods used in the problems of applied electromagnetics and wave propagation one may distinguish two methods most widely used in a VHF/UHF propagation in the atmospheric boundary layer, namely: split-step-Fourier and split-step Pade methods. Both methods are based on the parabolic approximation to a wave equation and differ in the method of obtaining the approximation of the exponential operator. Initially, the split-step parabolic approximation was introduced in a problem of applied acoustics where the wave propagation can be described by a scalar wave equation of elliptical type:
where u is the spectral amplitude of the wave component with frequency ω, k = ω/c is the wavenumber, c is the group velocity of the propagation and n is the refractive index of the medium and may vary with the coordinates, we assume, for now, a constant value of n, n = 1. Equation (26) is written for a two-dimensional case of the propagation since this case is the one most widely considered in applications of the split-step approximation.
Equation (26) can be factorised as follows:
representing the forward and backward propagating waves respectively. We may also remove fast phase variations with distance by introducing the slow-varying amplitude ũ, u = ũ exp (jkx). The truncated and factorised equation for amplitude ũ takes the form:
By retaining the only forward propagating wave we obtain the forward parabolic equation:
which has a formal solution:
where Δx is a range increment. As apparent from Eq. (30) the forward propagating wave at range x + Δx can be obtained from values of the wave amplitude at the previous distance x by applying the exponential operator in Eq. (30). In order to actually calculate the wave field, the exponential operator in Eq. (30) should be approximated in a form suitable for computations. Let us introduce a notation Z = 1/k2∂2/∂z2 and, depending on the type of approximation, we obtain three known types of the split-step approximation to parabolic equations:
a The standard or narrow angle approximation, obtained by expanding the square root in the exponent into a Taylor series and retaining the linear term:
This approximation is used in a split-step Fourier method to be discussed later, and is normally valid for narrow angles of propagation relative to axis x, not exceeding 15°–20°.
b The Claerbout approximation in the form:
which is reported to be valid for wider angles up to 30–40°. And finally:
c The split-step Pade approximation, when exponential is expanded into series:
where the coefficients αm, bm are determined numerically in the complex plane. The split-step Pade approximation is reported to be valid for angles of propagation relative to the x-axis of up to 90°.
All the above methods have been realized in a very powerful computational tech- nique well suited for parallel computing and widely used in numerous applications. With regard to the problem of radio wave propagation in the earth’s troposphere the most established computational approach is based on the split-step Fourier method equivalent to a standard narrow angle approximation of the square root in the expo- nential operator. Whilst the wide angle approximations, such as Claerbout and Pade approximations have been used in the problems of scattering on objects submerged in a free-space, the author does not know of any systematic derivation of the PadØ approximation in the case of radio wave propagation in a stratified troposphere over terrain or the sea boundary surface. We may notice that while the fundamental elliptic equation of type (22) can be obtained for potentials (Hertz functions) under free-space propagation conditions, the equations for the Debye’s potentials take a different form in the case of a stratified troposphere. In the presence of a random component of the refractive index the main equation for the slow varying envelope is given by Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and Applicationsthis equation.
Then we may introduce the operator:
and factorize Eq. (34) in a form similar to Eq. (25):
Then, following the above described procedure, we may retain the forward propagating wave obeying the truncated equation:
with formal solution at the marching step in the form:
The exponential can then be expanded into a series similar to Eq. (33) and the next range step solution is given by:
We may introduce the auxiliary function:
which can be found by multiplying both parts of Eq. (40) by 1+bmL and solving a second-order differential equation for fm. Substituting fm into Eq. (39) we have:
A somewhat modified approach to the implementation of the Pade method is to substitute a Pade series with a Pade product:
Now, we concentrate on the split-step Fourier method. Consider Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and Applicationsthis equation and assume the presence of deterministic and stratified refractivity in the medium, the random component δε is set to 0. Assume first that the propagation takes place in an unbounded medium, i. e., we have only requirements on the appropriate decay of the field at infinity. The next assumption normally employed in the split-step method applied to radio wave propagation in the troposphere is to assume the medium to be stratified over the z-coordinate, i. e. a stratified troposphere, thus removing the dependence on the y-coordinate and truncating Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and Applicationsthis equation to a two-dimensional (x, z) parabolic equation.
Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and ApplicationsEquation in this case can be presented in the form:
In a vertically stratified medium the formal solution to Eq. (42) is then given by:
where Δx is a range increment. Equation (43) forms the basis for a split-step Fourier implementation. Let us define a Fourier transform of the envelope W(x, z) as follows:
The inverse Fourier transform is given by:
where the subscripts e and o indicate that the field is even or odd, which, in turn, corresponds to horizontal or vertical polarization, respectively. The impedance boundary condition (Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and Applicationsthis equation) was also treated in Ref. where it was shown that such a boundary condition can be realized using a mixed Fourier transform.
where the term K(x) is defined as:
It is apparent that the implementation of the split-step Fourier algorithm employing the formulas (47) and (48) will be more complicated than the rather simple implementation for a perfectly conducting interface using formula (46). As shown in many other publications (we may refer to a practical implementation of the numerical methods in radio coverage prediction systems), the value of the implementation of the impedance boundary conditions becomes pronounced at vertical polarization at smaller distances and at lower VHF frequencies for long-range propagation, where the impact of the surface wave produced by the distributed images of the source provides a substantial contribution to the received field strength.
The important development in an implementation of the split step Fourier transform method is the capability to treat an irregular terrain profile, as reported in numerous publications. The approach was initially developed in underwater acoustics and then implemented in the studies of radio propagation in the troposphere.
Let us consider the field of horizontal polarization over an irregular terrain with the terrain profile given by the function of the distance T(x). (We may note that with |q| ≫ 1 the boundary condition is W(x, z = 0) = 0 for both polarizations.) The range dependent boundary condition is then given by W(x, z = T(x)) = 0. The approach is then to map the irregular terrain profile to a smooth or more flat one with consequent modifications of the wave equation. The mapping is made by introducing a change of variables. Let the new height and range variables be represented by:
In the new coordinate system the slow varying envelope W(x, z) can be written in the form:
where θ(χ, ς) is a phase correction term due to the irregular terrain. Substituting the new variables into Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and Applicationsthis equation and following the procedure we end up with the following equation for a new envelope:
and:
As observed, the implementation of the split-step Fourier transform can be done in a way similar to the case of a smooth terrain, the difference from Understanding Parabolic Approximation in Wave Propagation: Analytical Methods and Applicationsthis equation is the presence of the second derivative of the boundary interface, 2ςT′′(χ), in the exponential operator. Nonetheless, given that the split-step Fourier transform algorithm treats this term as a constant, the addition of the second derivative does not make a difference in the implementation though ensuring the important correction of the wave front due to scattering on the irregular terrain.
We may notice apparent similarities between the split-step Fourier method and the discrete implementation of the path integral method. In the case of propagation over the boundary interface, considered in Section 3***, implementation of the boundary conditions into the path integral approach will result in the appearance of the mirrored source, similar to the above formulas (47) and (48). In fact, numerical calculation of the discrete version of the path integral is practically realised by applying the fast-fourier transform algorithm to the exponential propagator at given step along the range x.
Finally, we can state that the above numerical methods constitute a powerful framework for quantitative analysis of radiowave propagation through the tropospheric boundary layer in the very general case of the refractivity condition. In particular, the methods described above can incorporate either non-uniformity of the refractivity profiles in the horizontal plane as well as in an irregular terrain. The limitation of the above methods, as in fact of all numerical methods, is that while they are quite helpful in a quantitative estimation of the propagation phenomena when the physical mechanism is clearly understood they lack the capability to provide a framework for qualitative analysis, especially in the case of combined effects of refraction and scattering on random irregularities of refractive index or a randomly rough sea surface.